The AT -{Transmuted - X} Family of Distributions

The Exponential Transmuted Exponential distribution (ETE) appeared in and in this paper we present a new generalization of the ETE distribution based on an application of the Ampadu-G family of distributions that appeared in [1,2]. We also show the new distribution is a good fit to some real-life data, indicating practical significance. As a further development, we propose a new class of distributions based on the structure of the weight function introduced in [3].

At first we recall the following definitions

Let λ > 0, ξ > 0 be a parameter vector all of whose entries are positive, and x ? R [2]. A random variable X will be said to follow the Ampadu-G family of distributions if the CDF is given by

And the PDF is given by

Where the baseline distribution has CDF G(x; ξ) and PDF g(x; ξ)

Assume the random variable T with support [0,∞) has CDF G(t; ξ) and PDF g(t; ξ) [2]. We say a random variable S is A_T - X(W) distributed of type II if the CDF can be expressed as either one of the following integrals

Where λ, ξ > 0, and the random variable X with parameter vector ω has CDF F(x; ω) and PDF f(x; ω)

As an application we consider the data on patients with breast cancer [1]. We assume the random variable T is exponentially distributed. So that the CDF of T is given by

For t, d > 0 and the PDF is given by


For t, d > 0. Now we recall the following, for some baseline distribution K(x), the transmuted family of distributions has CDF given by

Where b ? [-1, 1]. Now we assume X is transmuted exponentially distributed, so that if

For x, a > 0, then the CDF of X is given by

And the PDF is given by

Now from the first integral in 3.2 Definition, we have the following

The CDF of the A_{{Exponential}}-{Transmuted Exponential} distribution is given by

Where x, a, c, d > 0 and b ? [-1, 1]

If a random variable Q has CDF given by the above theorem, write 

The PDF of the AETE distribution can be obtained by differentiating the CDF
The AETE distribution is seen to be a good fit to real life data as shown in the figure 1.

 

Figure 1: The CDF of AETE(0.0133756, 0.216229, 3.93753, 0.910041) fitted to the empirical distribution of the data on patients with breast cancer [1].

Inspired by the structure of the weight function introduced in, we ask the reader to investigate some properties and applications of a so-called A_{{New T}} - X(W) family of distributions of type II [3]. We leave the reader with the following.

Assume the random variable T with support [0,∞) has CDF G(t; ξ) and PDF g(t; ξ). We say a random variable S_New is A_{{New T}} - X(W) distributed of type II if the CDF can be expressed as the following integral.

Where λ, ξ > 0, and the random variable X with parameter vector ω has CDF F(x; ω) and PDF f(x; ω)